A spatial random field model to characterize complexity in earthquake ...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B11, 2308, doi:10.1029/2001JB000588, 2002

A spatial random field model to characterize complexity in earthquake slip P. Martin Mai1 and Gregory C. Beroza Department of Geophysics, Stanford University, Stanford, California, USA Received 24 April 2001; revised 25 January 2002; accepted 30 January 2002; published 19 November 2002.

[1] Finite-fault source inversions reveal the spatial complexity of earthquake slip over the fault plane. We develop a stochastic characterization of earthquake slip complexity, based on published finite-source rupture models, in which we model the distribution of slip as a spatial random field. The model most consistent with the data follows a von Karman autocorrelation function (ACF) for which the correlation lengths a increase with source dimension. For earthquakes with large fault aspect ratios, we observe substantial differences of the correlation length in the along-strike (ax) and downdip (az) directions. Increasing correlation length with increasing magnitude can be understood using concepts of dynamic rupture propagation. The power spectrum of the slip distribution can also be well described with a power law decay (i.e., a fractal distribution) in which the fractal dimension D remains scale invariant, with a median value D = 2.29 ± 0.23, while the corner wave number kc, which is inversely proportional to source size, decreases with earthquake magnitude, accounting for larger ‘‘slip patches’’ for large-magnitude events. Our stochastic slip model can be used to generate realizations of scenario earthquakes for INDEX TERMS: 7209 Seismology: Earthquake dynamics near-source ground motion simulations. and mechanics; 7215 Seismology: Earthquake parameters; KEYWORDS: earthquake source characterization, complexity of earthquake slip, spatial random fields, correlation length of asperities growing with earthquake magnitude, earthquake rupture dynamics, strong ground motion simulation Citation: Mai, P. M., and G. C. Beroza, A spatial random field model to characterize complexity in earthquake slip, J. Geophys. Res., 107(B11), 2308, doi:10.1029/2001JB000588, 2002.

1. Introduction [2] Images of the spatial and temporal evolution of earthquake slip on fault planes provide compelling evidence that fault displacement is spatially variable at all resolvable scales. These finite-source rupture models are typically derived by inversion of low-pass-filtered strong ground motion recordings [e.g., Beroza and Spudich, 1988], sometimes augmented with teleseismic data [Hartzell and Heaton, 1983; Wald et al., 1991] and/or geodetic measurements [Heaton, 1982; Wald and Somerville, 1995; Yoshida et al., 1996]. Inversions of geodetic data only are also used to constrain larger-scale slip [Larsen et al., 1992; Bennett et al., 1995], and provide independent evidence for spatial variability of slip. Some studies of strong motion data indicate that the rupture velocity is spatially variable as well [Archuleta, 1984; Beroza and Spudich, 1988; Bouchon et al., 2000], adding another degree of complexity into the rupture process. This rupture variability is of great interest because it strongly influences the level and variability of damaging high-frequency seismic energy radiated by an earthquake [Madariaga, 1976, 1983; Spudich and Frazer, 1984]. 1 Now at Institute of Geophysics, ETH Hoenggerberg, CH-8093 Zurich, Switzerland.

Copyright 2002 by the American Geophysical Union. 0148-0227/02/2001JB000588$09.00

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[3] Numerous theoretical studies of extended-source earthquake models describe heterogeneous slip distributions on fault planes. Andrews [1980b, 1981] showed that a slip spectrum that decays as k
2 in the wave number domain leads to far-field displacements that follow the widely observed w
2 spectral decay. The fundamental assumption in this model is that stress drop,
s, is scale invariant, i.e., individual large and small earthquakes as well as subevents of different size have about the same stress drop. Based on this concept, Herrero and Bernard [1994] introduced the ‘‘k-square’’ model in which the slip spectrum decays as k
2 beyond the corner wave number, kc, which is related to fault length. In this representation, slip is fractal. The ‘‘k-square’’ model assumes that the rupture front propagates with constant rupture velocity vr , while the risetime depends on wave number. The resulting ground motions follow the w
2 decay for far-field displacements, and were used to study directivity effects [Bernard et al., 1996]. Another class of constant stress-drop, extended-source models are the composite source models in which the earthquake is composed of many small events of different size with a fractal size distribution, filling the rupture plane to form the main shock slip distribution [Frankel, 1991; Zeng et al., 1994]. The composite source model has been used successfully to simulate ground motions as well as to invert for earthquake slip distributions [Zeng et al., 1994; Zeng and Anderson, 2000; Su et al., 1996]. All of these models have in common a fractal slip distribution, and hence contain no

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Figure 1. Slip distributions from finite-source studies illustrate the spatial variability of slip on the rupture plane. The fault dimensions are plotted approximately on scale; gray scales indicate slip (in m). The slip models were derived by Cohee and Beroza [1994] (Landers), Sekiguchi et al. [2000] (Kobe), Wald et al. [1996] (Northridge), Archuleta [1984] (Imperial Valley), and Beroza and Spudich [1988] (Morgan Hill).

characteristic length scales to describe the size of asperities (large slip patches, areas of high stress drop). The only length scale in the fractal model is a ‘‘characteristic’’ source dimension, Lc (usually fault length) that determines the corner wave number, kc / 1/Lc, beyond which the spectra show power law decay. [4] In a recent study, Somerville et al. [1999] take a deterministic approach to correlate size and number of asperities with seismic moment for a set of finite-source rupture models. They find that the total number of asperities as well as the asperity size increases with seismic moment. The same study also indicates that for a few selected earthquakes the slip distributions may follow a k
2 decay in the wave number domain, but there has not yet been an attempt to rigorously verify the k
2 model for published

finite-source models. Somerville et al. [1999] propose that the correlation of size of asperity and number of asperities with seismic moment can be used to constrain simulated slip distributions that obey a k
2 spectral decay. [5] In this paper we propose a stochastic characterization of the spatial complexity of earthquake slip as found in finite-source slip inversions (Figure 1). We use a spatial random field model in which the slip distribution is described by an autocorrelation function (ACF) in space, or its power spectral density (PSD) in the wave number domain, each parameterized by characteristic length scales ax, az. We compile a database of 44 published finite-source rupture models for 24 different earthquakes (Table 1), and for each slip model we test whether a Gaussian (GS), an exponential (EX), or a von Karman (VK) ACF provides an

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Table 1. Source Parameters of Finite-Source Rupture Models Used in This Study No.

Location

Date

L

W

D

Mw

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Kanto San Fernando Tabas Coyote Lake Imperial Valley

9 January 1923 9 February 1971 16 September 1978 6 August 1979 15 October 1979

Borah Peak Morgan Hill

28 October 1983 24 April 1984

Michoacan Nahinni 1 Nahinni 2 N Palm Springs

19 September 1985 5 October 1985 23 December 1985 8 July 1986

Whittier Narrows Elmore Ranch Superstition Hills

10 October 1987 24 November 1987 24 November 1987

Loma Prieta

18 October 1989

Sierra Madre Joshua Tree

28 June 1991 23 April 1992

Landers

28 June 1992

Northridge

17 January 1994

Kobe

17 January 1995

Hyuga-Nada 1 Hyuga-Nada 2 Izmit (Turkey)

19 October 1996 2 December 1996 17 August 1999

Chi-Chi (Taiwan)

20 September 1999

130 19 95 10 35 42 42 52 30 27 180 40 48 22 22 10 25 24 20 40 38 40 40 7 35 22 84 80 78 77 17 20 18 18 64 60 60 60 32 29 135 125 85 84

70 19 45 10 13 11 10 26 10 11 140 17 21 15 15 10 10 10 12 14 17 20 14 6 20 20 18 15 15 15 26 26 21 24 20 16 20 20 32 29 20 22 22 42

2.01 1.35 0.38 0.23 0.39 0.62 0.41 0.39 0.26 0.17 1.46 0.56 0.50 0.15 0.13 0.26 0.28 1.17 0.83 1.29 1.06 1.14 1.80 0.16 0.11 0.12 1.39 2.00 2.47 1.99 0.71 1.07 1.00 0.73 0.35 0.66 0.82 0.71 0.57 0.46 3.26 1.52 2.36 3.93

7.80 6.76 7.10 5.87 6.46 6.59 6.45 6.78 6.23 6.10 8.01 6.68 6.76 6.10 6.06 5.91 6.19 6.61 6.48 6.87 6.85 6.94 6.96 5.51 6.23 6.12 7.18 7.22 7.27 7.20 6.63 6.79 6.68 6.63 6.73 6.83 6.96 6.91 6.81 6.69 7.59 7.38 7.39 7.72

appropriate description of the inferred slip distribution. We also examine the possibility of slip being fractal (meaning that there are no characteristic scale lengths, aside from the source dimensions) by estimating the fractal dimension, D. Finally, we analyze our measurements of correlation lengths ax, and az, and fractal dimension, D, for dependence on source parameters, i.e., whether scaling laws exist that relate the stochastic source parameters (fractal dimension, correlation lengths) to deterministic source parameters (fault length, fault width, seismic moment). [6] We first discuss the basic concepts of the spatial random field model, and outline the approach we take to determine the best fitting correlation lengths. We demonstrate the validity of the method and estimate its accuracy using simulated examples of heterogeneous slip maps. Then we apply the technique to published slip distributions, and examine the measured correlation lengths with respect to earthquake source parameters. We discuss the limitations of our analysis, and the implications of the results for rupture dynamics. We use our model to simulate slip distributions for hypothetical future earthquakes that

Mo 6.04E 1.61E 5.31E 7.56E 5.83E 9.03E 5.69E 1.79E 2.59E 1.70E 1.21E 1.26E 1.65E 1.68E 1.45E 8.67E 2.30E 9.30E 6.31E 2.39E 2.26E 3.01E 3.31E 2.21E 2.65E 1.80E 6.95E 7.92E 9.53E 7.59E 1.03E 1.85E 1.24E 1.04E 1.49E 2.10E 3.23E 2.80E 1.92E 1.27E 2.90E 1.38E 1.45E 4.57E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

20 19 19 17 18 18 18 19 18 18 21 19 19 18 18 17 18 18 18 19 19 19 19 17 18 18 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20 20 20 20

FM

Reference

RV RV RV SS SS SS SS N SS SS RV RV RV OB OB OB SS SS SS OB OB OB OB RV SS SS SS SS SS SS RV RV RV RV SS SS SS SS RV RV SS SS SS RV

Wald and Somerville [1995] Heaton [1982] Hartzell and Mendoza [1991] Liu and Helmberger [1983] Archuleta [1984] Hartzell and Heaton [1983] Zeng and Anderson [2000] Mendoza and Hartzell [1988] Beroza and Spudich [1988] Hartzell and Heaton [1986] Mendoza and Hartzell [1988] Hartzell et al. [1994] Hartzell et al. [1994] Hartzell [1989] Mendoza and Hartzell [1988] Hartzell and Iida [1990] Larsen et al. [1992] Larsen et al. [1992] Wald et al. [1990] Beroza [1991] Steidl and Archuleta, [1989] Wald et al. [1991] Zeng and Anderson [2000] Wald [1992] Bennet et al. [1995] Hough and Dreger [1995] Cohee and Beroza [1994] Cotton and Campillo [1995] Wald and Heaton [1994] Zeng and Anderson [2000] Dreger [1994] Hudnut et al. [1996] Wald et al. [1996] Zeng and Anderson [2000] Sekiguchi et al. [1996] Yoshida et al. [1996] Wald [1996] Zeng and Anderson [2000] Yagi and Kikuchi [1999] Yagi and Kikuchi [1999] Bouchon et al. [2000] Sekigucki and Iwata [2002] Yagi and Kikuchi [2000] Zeng and Anderson [2000]

can be used to calculate synthetic near-field ground motions.

2. Finite-Source Models as Spatial Random Fields [7] Spatial random field models are widely used in geosciences to describe quantities with nonhomogeneous spatial distribution [Goff and Jordan, 1988; Turcotte, 1989; Holliger and Levander, 1992]. They are characterized either in space by an ACF, C(r), or in the Fourier domain by a PSD, P(k), where r is distance and k is wave number. We consider three commonly used correlation functions, the Gaussian, exponential, and von Karman correlation function with the following expressions for C(r) and P(k): C ðr Þ 2

GS

e
r

EX

e
r

VK

GH ðrÞ GH ð0Þ

P ðk Þ ax az
14k 2 2 e

ð1Þ

ax az 3

ð1þk 2 Þ2 ax az ð1þk 2 ÞHþ1

:

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MAI AND BEROZA: RANDOM FIELD MODEL FOR EARTHQUAKE SLIP

Gaussian

Exponential

is characterized by a power law decay. In the twodimensional case, this can be written as [Voss, 1988] P ðk Þ /

1 1 / 4
D 2 k bþ1 kx þ kz2

ð2Þ

where D ¼ E þ 1
H: von Karman

In (3), D is the fractal dimension, E the Euclidean dimension of the fractal medium, therefore D = 3
H for a twodimensional fault plane. From (2) and (3) it follows that b = 1 + 2H. The ‘‘k-square’’ model therefore implies H = 1, or equivalently D = 2. [8] The differences among the random field models in (1) are illustrated in Figure 2 for four slip distributions with identical phasing. The bottom panel displays the spectral decay as a one-dimensional slice (at kz = 0) of the twodimensional spectrum. While the Gaussian random field is smooth with little short-scale variation, the remaining distributions show considerable short-scale variability. The power spectral densities illustrate these differences, with a very rapid decay for the Gaussian model, but a more gradual decay for the exponential, von Karman and fractal models. The beginning of the roll-off of the power spectra is related to the correlation lengths for the Gaussian, exponential, and von Karman ACF; for the fractal model, the PSD decays with a power law beyond a corner wave number, kc, which is related to the characteristic source dimension.

Fractal

Power Spectral Decay 0

GS EX VK FR

Power Spectral Density

10

1

10

ð3Þ

2

10

3. Estimating Spectral Decay Parameters 3

10

←k

c

4

10

1

10

0

10 Wavenumber (1/km)

1

10

Figure 2. Examples of the spatial random field models considered in this study, generated with identical phasing to facilitate the comparison. The grid dimensions are 30  30 km; the correlation lengths are a = 5 km for the Gaussian, exponential, and von Karman ACF. For the von Karman model, H = 0.8; for the fractal case, D = 2.2 and kc = 0.3. The bottom graph displays the corresponding power spectral decays of a one-dimensional slice (at kz = 0) of the two-dimensional power spectrum P(k).

where GH (r) = r HKH(r). In these expressions, H is the Hurst exponent, KH is the modified Bessel function of the first kind (order H ), and kx and kz are horizontal and vertical wave numbers, respectively. The characteristic scales are given by the correlation lengths in along-strike and direction, ffi downdip qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 z2 2 ax, az, respectively, and r ¼ a2 þ a2 ; k ¼ ax kx2 þ a2z kz2 . x z The Hurst exponent H in the expression for the von Karman ACF determines the spectral decay at high wave numbers. For H = 0.5 the von Karman ACF is identical to the exponential ACF. For a fractal medium, the power spectrum

[9] To estimate the best fitting correlation length for the Gaussian, exponential and von Karman correlation function, we use a grid-search algorithm that operates in the wave number domain by fitting power spectra for discrete values of correlation lengths a to measured spectral densities. The fractal dimension, D, can be estimated from a least squares fit to the power spectral decay beyond the corner wave number. [10] We examine average decay properties of the slip spectra using the Fractal Analysis from Circular Average (FACA) method [Anguiano et al., 1994], then we analyze the spectral decay in along-strike and downdip direction separately. In the FACA method, the fractal dimension of a two-dimensional fractal image is estimated from integrated spectral values along a radial wave number, kr , computed from the directional wave numbers, kx , kz , and hence only represents average properties of the random field. We extend this approach to more general random fields, parameterized by the ACFs given in (1). This method eliminates possible anisotropy, but it yields more stable average estimates for the decay parameters, which provide a starting point for the subsequent two-dimensional analysis of the slip spectra. We then estimate the best fitting correlation length ax, az, in along-strike and downdip direction, respectively, by iteratively sweeping through a large range of correlation lengths (from zero to the maximum source extent at a step size of 0.2 km). This is done for each direction separately using the one-dimensional slice at kz = 0

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Figure 3. Distribution of measured fractal dimensions, D, versus the input fractal dimension for simulated slip distributions. The left panel shows the results for unfiltered slip maps, and the right panel for wave number filtered models for which the two-dimensional spectra were low-pass filtered at 0.2 kny , where kny is the Nyquist wave number of the simulated models. Stars indicate the actual measurements, the triangle marks the median, and the crosses show the 1s bounds. The inset displays the cumulative distribution of all five populations (each with 20 realizations) with respect to each population’s median; the solid curve depicts the fit to a normal distribution with standard deviation s. The graphs indicate that the algorithm has difficulties recovering the fractal dimension for large D for the wave number filtered models. (kx = 0) in along-strike (downdip) direction of the twodimensional PSD. 3.1. Sensitivity of Spectral Decay Parameters [11] Before we apply the grid-search algorithm to the inferred slip maps we test the method on simulated slip distributions. We generate test models on a square grid of 30  30 km with sampling of dx = dz = 0.5 km, and input correlation lengths in the range of 5  a  20 km. For the von Karman ACF, we fix the Hurst number to H = 0.5, since this allows to simultaneously evaluate the performance of the exponential ACF. For the fractal model, we consider fractal dimensions in the range 2  D  3, where the lower limit characterizes the Euclidean dimension while the upper limit defines the ‘‘space-filling’’ (so-called Peano) twodimensional field. [12] To estimate the accuracy of the spectral decay parameters ax, az, H, and D, we apply the inversion to 20 simulated slip distributions for each correlation length and fractal dimension. The edges of these slip models are tapered to avoid unrealistically large displacements at the rupture boundaries. In a second step, we also low-pass filter the slip realizations to examine the effects that coarse spatial resolution and smoothing constraints in the slip inversion may have on the wave number spectrum and the resulting correlation length estimates. [13] Figure 3 shows that the inversion finds the fractal dimension for the unfiltered slip distributions accurately, and that the standard deviation for each population is about

equal. The deviation from each population’s input fractal dimensions for all slip realizations jointly is only s = 0.06. This changes for the wave number-filtered slip models (0  0.2 kny , where kny is the Nyquist wave number of the spatial grid). The cutoff wave number kf = 0.2 kny was chosen to mimic the spatial resolution for slip models with a grid size of dx 3 km (consistent with the average grid size used in slip inversions) and with spatial smoothing applied in the inversion. Different choices of kf change the D estimates slightly, but do not affect our conclusions. The standard deviation for each population is larger, and for fractal dimensions D 2.5 we underestimate the true value because the wave number filter removes the higher wave numbers required to represent small-scale variability accurately. The standard deviation for all estimates of D about each population’s median is s = 0.19. Based on these results we assume a D-dependent error for the estimated fractal dimension of inferred slip models, with s (D) increasing with D from s (D) = 0.1 for D 2 to s (D) = 0.25 for D 3. [14] In a similar manner we estimate the error in correlation length and Hurst exponent (Figures 4 and 5). For the unfiltered models the input values of a are accurately retrieved, with slightly increasing standard deviation for increasing correlation length a. The estimate for the Hurst exponent H is stable, with mean m = 0.49 and standard deviation s = 0.1 for an input value H = 0.5. The effect of the high wave numbers not being present in case of the filtered slip realizations can clearly be seen for short correlation lengths a that are consistently overestimated

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Figure 4. Same as Figure 3 for the correlation length, a, of the von Karman ACF for 20 slip realizations at each correlation length a = 5, 10, 15, and 20 km. The graph on the right shows that, for short correlation length, the algorithm is biased high for the wave number filtered slip models in which the high

by 2.75 (or 55%) for an input value of a = 5 km. For longer correlation lengths this effect decreases (though the spread within each population increases), and the error is on the order of ±2.6 (or 13%). We therefore assume an a-dependent error for estimated correlation lengths of inferred slip

models that decreases from 60% for short correlation lengths to 15% for large a. In case of the Hurst exponent H we find that its median estimate of mH = 0.57 for the filtered slip realizations is still close to the input value H = 0.5, indicating a slight bias toward higher values; the

Figure 5. Distribution of measured Hurst exponents, H, for an input value of H = 0.5, for unfiltered simulated slip models (left) and wave number filtered models (right). The solid line represents the fit of the measurements to a normal distribution with mean m and standard deviation s given in the top-left

2D-Power Spectral Density

Circular Averaged Spectral Density

-3

0

kny →

-2

-1 -2

-1

-3 -4 -

-

-5 -6

1

-7 -8

2

-9

Norm. Power Spectral Density

Wavenumber Down Dip [1/km]

10

0

-1

10

-2

10

← kc -3

10

c.a.PSD GS (a = 6.0 km) EX (a = 6.2 km) VK (a = 5.6 km, H = 0.66) FR (D = 2.42, kc = 0.185)

-4

10

Log (PDS)

3 -3

-2

-1

0

1

2

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MAI AND BEROZA: RANDOM FIELD MODEL FOR EARTHQUAKE SLIP

3

Wavenumber Along Strike [1/km]

-2

10

-1

10

0

10

C.A.Wavenumber [1/km]

Figure 6. FACA algorithm applied to the slip distribution of the 1979 Imperial Valley earthquake [Archuleta, 1984]. Vertical lines at kc and kny indicate the bandwidth of the spectral fit. The Gaussian model cannot fit the observed spectra; the fractal model requires a corner wave number, kc, and cannot capture the gradual roll-off. Both the exponential and the von Karman ACF provide an excellent fit to the averaged slip spectrum.

standard deviation increased to sH = 0.15. We adopt this value as the standard deviation for the estimates of the Hurst exponent for the inferred slip models. 3.2. Application to Finite-Source Models [15] We estimate the spectral decay parameters of 44 finite-source slip models (Table 1) that span a magnitude range of Mw = 5.9– 8.0 and comprise strike-slip and dip-slip earthquakes from various tectonic regimes. These slip models were derived using a number of inversion techniques and data sources (strong motion, teleseismic, geodetic, occasionally combined), different strategies to stabilize the inversion (regularization and smoothing constraints), and with spatial sampling as small as 0.5 km and as large as 10 km. In order to obtain a uniform wave number representation for all models and to facilitate the comparison among slip models, we bilinearly interpolate all models onto 1  1 km grid spacing such that the power spectral decays are directly comparable. We transform the interpolated models into the two-dimensional wave number domain using zeropadded grids of size 256  256 km so that each model has an equal number of data points for the inversion. [16] First we apply the FACA algorithm to estimate the average spectral decay parameters of the slip maps. Figure 6 displays an example of this method, applied to the slip model for the 1979 Imperial Valley earthquake [Archuleta, 1984]. We use the circular averaged PSD over the entire wave number range to estimate the averaged spectral decay parameters for the Gaussian, exponential and von Karman ACF, while the fractal dimension is computed from the power spectral decay beyond the corner wave number, kc. The results shown in Figure 6 already exhibit some important properties: the Gaussian ACF is not a viable model to describe heterogeneous slip in earthquakes, but the expo-

nential and von Karman ACF provide an accurate description of the spectral decay. Both models capture the gradual roll-off at low wave number as well as the high wave number decay. The fractal model, on the other hand, is a bilinear model in log-log space (flat for k < kc, linearly decaying for k kc), and does not capture the gradual rolloff of the observed spectrum. [17] To estimate the directional dependence of the spectral decay parameters, we apply a grid-search algorithm to one-dimensional slices in the along-strike and downdip directions of the two-dimensional PSD. Figure 7 displays the results for the 1979 Imperial Valley model for which slip was found to have occurred in an horizontally extended zone at depth [Archuleta, 1984] (see also Figure 1). The inversion recovers this anisotropy with correlation length in along strike direction, ax, being much larger than in downdip direction, az. The along-strike Hurst exponent H is smaller that the downdip Hurst exponent, indicating that a single fractal dimension (D = 3
H ) may not be sufficient to model heterogeneous slip distributions. Although the exponential and the von Karman ACF measure very similar correlation lengths (and hence RMS misfits) for this particular example, we generally observe that the von Karman ACF returns slightly lower RMS misfits than the exponential ACF due to the additional free parameter (Hurst exponent H ) in fitting the spectral decay. 3.3. Resolution of Spectral Decay Parameters [18] Before we discuss the estimates of correlation length and fractal dimension for the slip distributions in Table 1, we want to point out that the slip-inversion process exerts a strong influence on the measurements of lengths scales (asperity sizes) or fractal dimensions of finite-source rupture models, using either the stochastic approach presented in

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Figure 7. Fit of correlation functions to the power spectral decay of the slip distribution of the 1979 Imperial Valley earthquake [Archuleta, 1984]. Vertical lines at kc and kny indicate the bandwidth of the spectral fit. Note the difference in the correlation lengths in along-strike and downdip directions for the von Karman and the exponential ACF, reflecting the elongated nature of the area of large slip in this particular model (see Figure 1).

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Table 2. Estimates of Fractal Dimension, D, Correlation Lengths, a, and Hurst Exponents, H, for Slip Models Listed in Table 1 Fractal

Exponential

von Karman

No.

Leff

Weff

kc

kny

D

a

ax

az

a

ax

az

H

Hx

Hz

1 2 3 4a 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25a 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

92.1 10.9 64.5 5.0 23.5 25.4 21.7 32.6 20.3 14.1 134.9 21.7 24.2 15.9 14.9 7.6 16.2 17.0 13.2 25.3 24.4 31.3 26.3 3.3 15.0 9.8 43.3 56.4 47.0 50.5 11.1 15.6 13.6 11.6 47.9 38.1 39.4 35.6 21.6 21.1 95.5 83.6 57.3 53.0

52.4 10.9 30.5 5.2 7.5 8.6 6.3 20.1 6.6 8.9 96.8 10.1 11.6 10.2 10.9 6.9 6.3 7.6 7.2 9.0 10.6 12.1 9.5 3.1 12.5 9.8 13.2 13.5 11.5 10.6 17.5 17.7 14.8 14.2 17.1 15.5 13.6 13.7 23.6 21.1 16.4 18.4 14.3 26.6

0.030 0.183 0.048 0.392 0.175 0.156 0.206 0.080 0.201 0.183 0.018 0.145 0.128 0.161 0.159 0.275 0.219 0.190 0.215 0.151 0.135 0.115 0.143 0.630 0.146 0.203 0.099 0.092 0.108 0.115 0.147 0.120 0.141 0.156 0.079 0.091 0.099 0.101 0.088 0.095 0.071 0.066 0.088 0.056

0.276 1.571 0.697 1.571 2.199 1.112 1.571 0.927 3.142 1.571 0.314 1.492 1.257 1.439 1.439 2.001 1.414 1.414 1.571 3.142 1.524 1.100 1.571 3.142 3.142 3.142 0.861 0.839 1.935 1.571 3.142 1.561 2.271 1.571 1.530 1.257 0.707 1.571 1.074 1.074 1.571 1.065 0.873 0.785

1.89 1.79 2.19 1.50 2.25 2.18 2.26 2.10 2.78 3.04 2.27 2.64 2.54 2.52 2.37 2.24 1.92 2.01 2.45 2.29 2.23 2.34 2.21 2.22 1.50 1.99 2.20 2.44 2.33 2.28 2.29 2.17 2.41 2.43 2.40 2.42 2.60 2.56 2.19 2.28 2.37 2.52 2.12 2.41

32.6 5.2 19.4 1.8 6.2 6.6 4.6 11.4 5.0 5.2 50.0 6.2 6.6 5.8 5.2 3.2 4.8 5.4 4.2 6.6 7.2 8.2 6.8 1.2 6.4 3.0 10.2 12.4 10.6 10.6 5.2 8.2 6.6 5.6 12.8 11.6 10.0 9.2 10.0 10.0 18.8 17.2 13.4 15.8

43.6 5.2 23.0 1.8 11.2 10.8 7.2 13.8 9.8 6.6 57.4 7.2 7.8 7.0 6.4 3.6 7.2 7.4 5.6 10.0 10.8 13.4 11.4 1.2 6.4 3.0 14.2 29.0 19.8 24.2 5.0 6.8 6.6 4.8 17.6 15.6 17.2 13.2 9.2 10.6 34.8 33.0 24.4 22.0

22.6 5.0 12.2 1.8 3.2 3.8 2.6 9.2 3.0 4.0 38.6 4.8 4.8 4.6 4.6 3.0 2.8 3.6 3.0 4.0 4.2 5.2 4.0 1.0 6.4 3.6 6.0 5.8 4.8 5.0 7.4 9.2 7.0 6.0 7.4 7.2 5.8 5.8 10.6 10.6 6.8 7.8 6.0 10.2

24.4 4.0 15.8 1.2 5.2 5.4 3.8 9.0 6.4 7.2 41.4 6.0 6.6 5.0 4.8 2.8 3.8 4.0 4.2 5.6 6.2 6.4 6.0 0.8 3.8 2.2 8.4 10.2 9.6 10.4 4.6 6.8 5.2 5.2 11.6 10.6 9.6 10.8 8.2 7.6 16.8 17.6 10.6 14.4

32.6 4.0 18.4 1.2 10.4 9.0 6.6 10.8 12.6 10.6 42.2 7.8 6.8 6.4 5.6 3.2 5.6 8.4 6.0 10.0 10.0 11.6 9.8 1.0 3.8 2.4 11.8 24.4 17.0 17.6 4.0 6.0 5.6 4.2 19.2 17.0 16.8 18.0 7.4 8.6 31.2 33.8 18.4 19.8

17.4 3.8 10.2 1.2 2.8 3.4 2.2 7.4 3.0 4.8 34.8 5.0 5.4 3.8 3.8 3.4 2.2 2.8 2.6 3.4 3.8 4.6 3.8 1.0 3.8 2.6 5.0 5.0 5.0 4.4 7.0 7.4 5.8 4.8 7.0 8.6 5.4 6.2 9.0 9.8 6.4 7.4 4.8 9.4

1.11 1.21 0.78 1.50 0.80 0.82 0.78 0.90 0.22 0.10 0.74 0.52 0.50 0.82 0.62 0.76 1.08 0.99 0.48 0.74 0.70 0.88 0.68 0.78 1.50 1.01 0.80 0.80 0.62 0.54 0.68 0.83 0.84 0.66 0.62 0.62 0.61 0.33 0.81 1.02 0.63 0.47 0.84 0.64

1.11 1.21 0.81 1.50 0.60 0.81 0.66 0.90 0.21 0.00 0.92 0.50 0.70 0.68 0.77 0.70 1.08 0.32 0.41 0.50 0.59 0.70 0.72 0.78 1.50 1.01 0.80 0.80 0.70 0.72 0.82 0.83 0.82 0.73 0.40 0.43 0.56 0.22 0.87 0.82 0.63 0.47 0.91 0.66

1.11 1.21 0.81 1.50 0.75 0.86 1.02 0.90 0.48 0.26 0.64 0.46 0.34 0.89 0.80 1.15 1.08 0.99 0.86 0.71 0.75 0.70 0.62 0.78 1.50 1.01 0.80 0.80 0.65 0.72 0.58 0.83 0.82 0.86 0.58 0.22 0.72 0.35 0.82 0.65 0.60 0.59 0.94 0.62

a

These earthquakes cannot be described with the von Karman ACF or fractal model but rather with a Gaussian ACF.

this paper or a deterministic approach [Somerville et al., 1999]. The results strongly depend on the limitations of finite-source inversion studies. In these inversions, the rupture plane is discretized into many subfaults where the size of the subfaults may depend on the rupture dimensions, on the availability, spatial distribution and frequency bandwidth of data, and the inversion method used. The frequency range of the data used in the inversion is band-limited to the low frequencies, ranging from 0 Hz in geodetic inversions to a maximum of 4 Hz in inversions of strong motion data. These data limitations (in spatial distribution, frequency, and quality) leave any slip inversion with a component of the model that cannot be constrained by the data (the null space). Also, the resolution of the original models of slip may be anisotropic due to gridding or geometric effects. These factors taken together impose strong limitations on the range of spatial wave numbers present in the inferred slip distributions. [19] We limit the largest wave number to be considered in the inversion to the Nyquist wave number, kny = p/dx (dx is

the spatial discretization in the fault slip inversion). Since the spatial sampling in the slip inversion depends on the data type and their distribution, as well as the frequency range of seismic recordings, the limiting Nyquist wave number implicitly accounts for these factors in the spectral analysis. Table 2 lists kny , as well as the corner wave number kc which is needed to determine the fractal dimension, D. The bandwidth for the spectral fitting (0  k  kny for the ACFs, kc  k  kny for the fractal model) of all slip models is depicted in Figure 8, showing that kc decreases for larger magnitudes (i.e., larger dimensions); kny exhibits a similar trend, though more erratic. The usable wave number range for the two largest earthquakes in our data set (no. 1, the 1923 Kanto earthquake, no. 11, the 1985 Michoacan earthquake) is extremely narrow, and hence the estimates of correlation length and fractal dimension for these events may not be very reliable. [20] Finite-source inversions also include smoothing constraints to stabilize the inversion. The amount of smoothing, however, is often not a clear-cut objective choice, but rather

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MAI AND BEROZA: RANDOM FIELD MODEL FOR EARTHQUAKE SLIP

Figure 8. Approximate bandwidth of spectral fitting for all slip models in Table 1. The corner wave number kc is obtained from the effective source dimension; the Nyquist wave number kny is computed from the spatial discretization used in the fault slip inversion. Values of kc and kny are listed in Table 2. a subjective one to avoid oscillatory slip maps. We explore this effect using a set of inversion results with different levels of smoothing, made available to us by Sekiguchi and Iwata [2002], who used Akaike’s Bayesian information criterion (ABIC) [Akaike, 1980] to determine the optimal amount of smoothing in the inversion. Here, the smoothing parameter l is varied from less damping (l = 0.01, data ‘‘overfitted,’’ oscillatory/heterogeneous solution) to more damping (l = 0.5, data ‘‘underfitted,’’ very smooth solution). Analyzing the six inversions for varying l (Figure 9), we find that fractal dimension D decreases slightly with increasing l, while the Hurst exponent H increases with increasing l, demonstrating that more smoothing in the inversion results in faster spectral decay, and therefore less short-scale spatial variability in the slip distribution. The

Correlation Length Correlation Length a [km]

Fractal Dimension D

3 2.8 2.6 2.4 2.2 2

-2

10

Smoothing Parameter λ

0

10

30

Hurst Exponent 1

a ax az

25

Hurst Exponent H

Fractal Dimension

effect of l on the estimates of D and H, however is small (less than 10%). Figure 9 shows that the amount of smoothing had a stronger effect on the estimates of the correlation lengths, particularly in the along-strike direction in which ax increases from 16 km to 24 km with increasing l (a 50% increase). The effect is smaller for the circular-averaged correlation length (30% increase), and negligible in the downdip direction (5% increase). [21] Measurements of asperity size, characteristic scale lengths or spectral decay parameters, based on inferred finite-source models, are therefore only estimates of the true slip complexity during an earthquakes. Nevertheless, we believe that the characteristic scales we find from these slip models are likely to be representative of future earthquakes. In particular, since they are themselves derived

20 15 10 5 0

-2

10

Smoothing Parameter λ

0

10

H Hx Hz

0.8 0.6 0.4 0.2 0

-2

10

Smoothing Parameter λ

Figure 9. Estimates of fractal dimension D (left), correlation length a (center), and Hurst exponent H (right) for six slip inversions with variable damping, l, for the 1995 Kobe earthquake [Sekiguchi et al., 2000]. See text for details.

0

10

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Figure 10. Fractal dimension, D, measured for 44 slip models given in Table 1. There is no evidence that D is related to magnitude or faulting style. The normal probability plot (inset) indicates that the measurements are normally distributed, with median m = 2.29 and standard deviation s = 0.23.

from strong motion data, they can be used to simulate complex earthquake slip for strong motion prediction.

4. Spectral Decay Parameters of Finite-Source Models [22] In the following we discuss the estimates of the spectral decay parameters (fractal dimension, D, Hurst exponent, H, and correlation lengths a, ax and az) for 44 published finite-source rupture models (Table 1). The results are listed in Table 2 for each slip model, along with the effective source dimensions, Leff and Weff [Mai and Beroza, 2000]. We use these measurements to examine possible dependencies of the spectral decay parameters on source parameters (moment magnitude, Mw , fault length, Leff, fault width, Weff). 4.1. Fractal Dimension D [23] Although fractal dimension D and Hurst exponent H are related by (3), they are parameters of two different random field models, determined by two different strategies. H is estimated within the grid-search algorithm, D is computed within the FACA algorithm. We find, however, generally good correspondence between D and H estimates, approximately satisfying (3). [24] The estimates of fractal dimension D are shown in Figure 10, with 1s error bars s (D) as outlined above. The inset shows a quantile plot of all measurements, indicating that the values of D are normally distributed. We find no

dependence of fractal dimension on moment magnitude or source dimensions, and there are no significant differences between strike-slip or dip-slip earthquakes. We find a median value of D = 2.29 (mean = 2.31, sD = 0.23). These values are also robust applying a boot-strapping technique. t-test statistics indicate that the median estimate for D is not significantly different from D = 2 at the 95% level, as proposed in the constant stress-drop model [Andrews, 1980b; Herrero and Bernard, 1994]. For D = 2.29, however, the slip spectrum decays less rapidly than for D = 2, and hence leads to more short-scale variability of slip than the ‘‘k-square’’ model. Assuming that D > 2 is not an artifact of the fault slip inversion method or the inversion for the spectral decay parameters, the observation that D

2 has multiple possible interpretations: (1) nonconstant stress drop for subevents in a cascade earthquake source model [Frankel, 1991], (2) variability in risetime and/or rupture velocity is mapped into a more heterogeneous slip distribution [Herrero and Bernard, 1994], and (3) surfical geometrical complexity of fault systems extends to depth [Okubo and Aki, 1987; Aviles et al., 1987]. [25] For an earthquake source model consisting of a cascade of ruptures of different sizes the average stress drop over an area on the fault is assumed to be proportional to the standard deviation of the spatial variations in stress drop over that area [Frankel, 1991]. Then, the stress drop over some rupture zone with radius R will be
s / Rh [Frankel, 1991]. For h = 0, stress drop is independent of

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Figure 11. Hurst exponent, H, measured for 44 slip models listed in Table 1. We find no evidence that H is related to magnitude or faulting style. The distribution in each panel is normal, with median (m) and standard deviations (s) given in the top-right corner.

scale length, and all subevents will have the same stress drop, independent of their size (magnitude). In contrast, D > 2 implies that h < 0 and hence stress drop
s increases as the rupture size becomes smaller (smaller length scales). Although decreasing stress drop with increasing magnitude is not generally observed, it is in qualitative agreement with a study on strength of asperities [Sammis et al., 1999]. [26] An alternative interpretation for D > 2 is that it is an artifact that arises from finite-source inversion methods that do not accurately recover the true variability in risetime or rupture velocity. Although dynamic rupture modeling indicates that both risetime and rupture speed are spatially variable over the fault plane [Day, 1982], finite-source inversion often treat them as being constant or slowly varying. In order to fit the data, the inversion may therefore map variability in risetime or rupture velocity into a more heterogeneous slip distribution. [27] Independent evidence for D > 2 is found in studies on the geometrical complexity of fault systems [Okubo and Aki, 1987; Aviles et al., 1987]. In particular, Okubo and Aki [1987] find D = 1.31 ± 0.02 for the mapped fault trace of the San Andreas Fault which translates into D = 2.31 for the fault surface using Dsurface = Dtrace +1 [Okubo and Aki, 1987], consistent with our measurement of D = 2.29. In our opinion, the observation of D > 2 is probably attributable to a combination of incorrect mapping of rupture variability into the slip distribution as well as the geometric irregularity of the fault surface that is not accounted for in slip inversions. Whether the geometric complexities cause the rupture variability, or whether these are separate effects, remains to be addressed. 4.2. Hurst Exponent H [28] The inversion for spectral decay parameters returns three estimates of the Hurst exponent for each earthquake (Figure 11), an estimate for the circular-averaged spectrum (H, left), for the along-strike (Hx, center) and for the downdip direction (Hz, right). The median values m and standard deviations s (top right corner in each panel) for the three cases are close to each other, indicating that there is no strong directional dependence of H, and that the Hurst

exponent can be reliably estimated from the slip spectra. We find no evidence that the Hurst exponent depends on magnitude. It is interesting to note that the H estimates are close to H = 0.5 for which the von Karman ACF is identical to the exponential ACF, explaining why both the exponential and the von Karman ACF provide similar fits to the observed spectral decay, and generally result in similar correlation length estimates for a given slip model (Table 2). The median estimates for H and D approximately satisfy, D = 3
H, although they were derived using different approaches. H 0.75 may again indicate that the constant stress drop model (D = 2, or equivalently, H = 1) [Andrews, 1980b; Herrero and Bernard, 1994] may not apply to the published slip maps. The observation that H > 0.5 becomes important when simulating slip distributions under the condition of finite static self-energy (see Appendix A). 4.3. Correlation Lengths for Finite-Source Models [29] The correlation length estimates for the exponential and the von Karman ACF are listed in Table 2; the Gaussian ACF is not discussed further because it does not provide an adequate fit to the observed spectra. In the following discussion, we will focus on the correlation length estimates for the von Karman ACF because (1) the Hurst exponents are found to be close to H = 0.5 for which the von Karman ACF is identical to the exponential ACF, leading to similar correlation lengths estimates for the two ACFs and our conclusions hold for either ACF, and (2) we found that the von Karman ACF generally provides a better fit to the observed spectral decay than the exponential ACF or the fractal model, and we therefore adopt the von Karman ACF as our preferred model to describe complexity of earthquake slip. [30] The correlation lengths for all 44 slip models are listed in Table 2. It is interesting to note that the measurements of correlation lengths (but also D and H ) show a remarkable consistency for those earthquakes for which multiple finite-source studies exist. Among the best studied earthquakes are the 1989 Loma Prieta earthquake (nos. 20– 23) and the 1995 Kobe event (nos. 35– 38), for which we have four slip models each. For Loma Prieta, for instance, D is about 2.23 with little scatter, and the (von Karman)

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Figure 12. Measured correlation length, a, versus effective fault length, Leff, (top row) and fault width, Weff, (bottom row) for 44 slip models listed in Table 1. The effective source dimensions are given in Table 2, calculated following the work of Mai and Beroza [2000]. There is clear evidence that the correlation length a increases with increasing source dimensions. The graphs also show regression curves (solid lines; broken lines indicate 95% confidence limits) with intercept b0, slope b1, and correlation coefficient r2 (top-left corner). Regression coefficients for strike-slip and dip-slip earthquakes considered separately are listed in Table 3. estimates for a, ax, and az are 5.8 (±0.4), 10.8 (±1.2), and 3.8 (±0.3), respectively. We obtain similar consistency in case of the Kobe earthquake for the fractal dimension (D = 2.56 ± 0.09) and the correlation lengths (a = 10.6 ± 1.5, ax = 20.1 ± 2.0, and az = 7.4 ± 0.9). The variation of the correlation length estimates among these models is only about 10%. This is even more remarkable because these source models were derived using different data sources and inversion techniques, and to some extent the resulting slip distributions are dissimilar. In contrast, the correlation length estimates for the 1992 Landers earthquake (nos. 27– 30) differ by about a factor of two in the along-strike direction, reflecting the more localized slip in the models by Cohee and Beroza [1994] and Wald and Heaton [1994], compared to the solutions by Cotton and Campillo [1995] and Zeng and Anderson [2000]. Generally, the statistical properties of earthquakes with multiple finite-source slip models seem to be recovered in a consistent manner, and the random field model we propose is able to capture this similarity. [31] The correlation lengths we obtain are also similar to the asperity dimensions found by Somerville et al. [1999]. While we have estimated stochastic properties of a slip

distribution, their measurements are deterministic, and hence the two sets of measurements are not necessarily directly comparable. We find, however, that for slip models with multiple, small asperities [Somerville et al., 1999], we obtain short correlation lengths and low Hurst exponents. Likewise, where Somerville et al. [1999] find only one or two larger slip patches, we observe longer correlation lengths and larger Hurst exponents. In the deterministic model, the asperity size increases with source dimension (magnitude) [Somerville et al., 1999], an observation that is confirmed by our measurements in that larger earthquakes tend to have longer correlation lengths. We therefore conclude that the ‘‘characteristic length scales’’ measured in this study agree qualitatively with those by Somerville et al. [1999], though ours is a stochastic representation of slip complexity, whereas theirs is a deterministic one. 4.4. Scaling of Correlation Lengths With Source Dimensions [32] A careful analysis of the correlation lengths in Table 2 suggests that a, ax, and az depend on source dimension. This is manifest in Figure 12, which displays the measured

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Table 3. Coefficients for the Scaling of the von Karman Correlation Lengths a With Source Parameters Moment Magnitude, Mw, Fault Length, Leff, and Fault Width, Weff, Separated With Respect to Faulting Style FMa

log(a) = b0 + b1Mw

SS DS AL SS DS AL SS DS AL

0.51 0.46 0.49 0.17 0.28 0.24 0.95 0.43 0.42

(0.07) (0.06) (0.04) (0.01) (0.01) (0.01) (0.10) (0.02) (0.03)

SS DS AL SS DS AL SS DS AL

0.59 0.49 0.53 0.34 0.31 0.33 1.74 0.45 0.42

(0.08) (0.06) (0.06) (0.03) (0.01) (0.01) (0.25) (0.04) (0.07)

SS DS AL SS DS AL SS DS AL

0.33 0.38 0.37 0.06 0.21 0.15 0.44 0.35 0.35

(0.06) (0.07) (0.05) (0.01) (0.02) (0.02) (0.04) (0.01) (0.01)

a = b0 + b1Leff a = b0 + b1Weff

log(a) = b0 + b1Mw a = b0 + b1Leff a = b0 + b1Weff

Intercept (Error), b0 (sb0)

Standard Deviation, s

Correlation Coefficient, r2

Circular Average
2.62 (0.47)
2.30 (0.39)
2.46 (0.30) 1.78 (0.61) 0.34 (0.52) 0.59 (0.59)
2.29 (1.21) 0.38 (0.52) 2.07 (0.62)

0.17 0.13 0.15 1.52 1.64 2.39 1.95 1.64 2.78

0.74 0.78 0.75 0.89 0.97 0.88 0.81 0.97 0.84

Along-Strike
2.93 (0.54)
2.43 (0.44)
2.60 (0.38) 1.86 (1.12) 1.10 (0.40) 1.54 (0.57)
4.87 (2.86) 1.83 (1.23) 6.07 (1.53)

0.19 0.15 0.19 2.79 1.27 2.31 4.61 3.88 6.80

0.74 0.76 0.69 0.90 0.98 0.94 0.73 0.85 0.47

0.15 0.17 0.18 1.41 2.87 3.61 0.76 0.76 0.78

0.59 0.58 0.54 0.52 0.84 0.56 0.86 0.99 0.98

Slope (Error), b1 (sb1)

Equation

Downdip log(a) = b0 + b1Mw a = b0 + b1Leff a = b0 + b1Weff

a


1.62 (0.42)
1.79 (0.51)
1.80 (0.37) 2.25 (0.57) 1.12 (0.91) 1.07 (0.89)
0.39 (0.47) 0.58 (0.24) 0.55 (0.18)

SS, strikeslip; DS, dipslip; AL, all mechanisms.

correlation lengths for all slip models in Table 2 with respect to effective source dimensions [Mai and Beroza, 2000], fault length Leff (top) and fault width Weff (bottom). Error bars are computed based on the sensitivity tests for random slip models (see earlier section). The graphs demonstrate that the circular-averaged along-strike and downdip correlation lengths increase with source size. Linear regressions of correlation lengths on source dimension show this dependence with most correlation coefficients of r2 > 0.80. The downdip correlation lengths az depend only mildly on fault length (r2 = 0.56), likewise for the dependence of the along-strike correlation lengths ax with fault width (r2 = 0.47). The regression coefficients for the intercept (b0) and the slope (b1) are given in the upper left corner; Table 3 lists faulting style-dependent coefficients. [33] The regression of ax on fault length Leff and az on fault width Weff can be used to establish simple scaling relations between these source parameters, which in turn are useful to estimate the correlation length for future earthquakes. The results shown in Figure 12 and Table 3 suggest a simplified scaling as ax 2:0 þ 13 Leff az 1:0 þ 13 Weff

ð4Þ

in which the slope of b1 = 13 in (4) indicates that correlation lengths scale linearly with effective source dimension. For earthquakes with small aspect ratio Leff/Weff, (4) yields

roughly isotropic correlation lengths, while for earthquakes with large aspect ratios (i.e., great strike-slip earthquakes) the correlation length along-strike is larger; (4) also implies the ratio ax/Leff 1/3, which is represented in Figure 13. For dip-slip earthquakes, this ratio remains constant over the given magnitude range, while for strike-slip earthquakes the ratio increases with increasing magnitude, with az/Weff perhaps saturating at Mw 7. For all three cases (circularaveraged along-strike and downdip), the ratio is 0.38 (±0.09), varying in the narrow range of 0.25– 0.6. [34] Figure 14 shows log(a) versus moment magnitude, Mw, again overlain by least squares regression curves. In case of circular-averaged (a) or along-strike (ax) correlation length, the slope of the regression is about 0.5, indicating that the correlation length increases with magnitude in a self-similar fashion. Although the estimates in downdip direction deviate from this apparent self-similarity, we hypothesize a simplified relation between the correlation length and moment magnitude (Table 3) as   log ag
2:5 þ 12 Mw logðaz Þ
1:5 þ 13 Mw

ð5Þ

where ag is either circular-averaged or along-strike correlation length. The relations in (5) are very similar to results of a recent study in which the characteristic subevent size
l was estimated based on source-parameter modeling [Beres-

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pffiffiffiffiffiffiffiffi Figure 13. Ratio of correlation lengths and effective source dimensions, a= Aeff (left), ax/Leff (center) and az/Weff (right), with the medians (st.dev.) given for the different faulting styles (ss: strike-slip; ds: dipslip; al: all mechanisms). The ratios are independent of magnitude and tend to be in the narrow range of 0.25– 0.6; on average, the correlation lengths are about 40% of the characteristic source dimension.

nev and Atkinson, 2001]. They find that
l scales with magnitude as logð
l Þ ¼
2 þ 0:4 Mw ;

ð6Þ

and reanalyzing the catalog of events of Somerville et al. [1999] they find logð
l Þ ¼
2 þ 0:5 Mw ;

ð7Þ

interpreting
l as the asperity size. The simple linear scaling presented in (5), (6), and (7) for the correlation length, subfault size and asperity size, respectively, suggest a fundamental property of extended-source earthquake models, namely, that the characteristic length scales with earthquake magnitude. This property can be used to estimate the characteristics of hypothetical future earth-

quakes, but it also may be interpreted in terms of the rupture physics.

5. Discussion [35] In this section we address the question how well the fractal dimension D may be resolved from inversion of near-source strong motion data, and what effect variable D has on synthetic seismograms. We propose what additional data may be useful to constrain the fractal dimension or correlation lengths of earthquake slip, and we discuss the implications of our results for dynamic rupture propagation. 5.1. Data Dependence of D Estimates [36] The fractal dimension obtained from analysis of finite-source models is subject to several sources of uncertainty. We are encouraged that our results are meaningful

Figure 14. Measured correlation length, a, versus moment magnitude, Mw , for 44 slip models listed in Table 1; filled squares denote strike-slip earthquakes and open diamonds represent dip-slip events. The slope of the regression curves (solid lines) is b1 0.5, an indication that correlation lengths scale with moment magnitude in a self-similar manner. Table 3 lists the entire set of regression coefficients.

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Figure 15. Phase-identical slip distributions with variable fractal dimension, D, with hypocenters indicated by a star (left), for which we calculated near-source synthetic seismograms (center). Displayed are the fault-parallel components, bandpassed at 0.01– 2 Hz. Note how the variation in D is reflected in the waveforms and the corresponding peak-ground velocities. The velocity spectra (right, vertically shifted) display only minor differences, and their decay (/ w
1) is consistent with observed flat acceleration spectra.

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because we find that fractal dimension D or correlation length a (Table 2) are very consistent for independently derived models of the same earthquake. The spatial discretization of the source models is potentially important, but Figures 8 and 10 suggest that it exerts only a minor influence on the estimate of D, i.e., the fractal dimension does not depend significantly on the grid size. Earthquakes with Mw 7 generally have a lower Nyquist wave number (kny), yet their D estimates agree with the general observation, D > 2. Only for the 1923 Kanto earthquake (no. 1) do we find a considerably smaller fractal dimension (D = 1.89), attributable to the sparse historic data for this event. [37] In contrast, we observe a significant influence of the type of data used in the fault slip inversion. Inversions of only geodetic data result in rather low estimates of D (event nos. 17, 18, 25, and 32). Likewise, models derived from teleseismic data alone have fractal dimension that is somewhat below the average D (event nos. 39, 40, and 43). Using or incorporating strong motion data in the slip inversion, on the other hand, results in a larger degree of spatial complexity (i.e., larger D). This is attributable to the sensitivity of near-source strong motion recordings to the details of the rupture process. [38] To investigate the effect that variable D exerts on near-source seismograms, we generate four slip models (Mw = 7, source dimensions 15  40 km) with varying fractal dimension, but identical phase, such that asperities are located in the same regions (Figure 15, left column). Using the same hypocenter, average rupture velocity, risetime and slip-velocity function, we calculate synthetic seismograms on a grid of 36 locations. The seismograms in Figure 15 display waveforms for an observer located at x = 45 km along-strike at a distance of 5 km in the fault-normal direction. We band-passed the seismograms from 0.01 to 2 Hz to reflect the bandwidth of seismic data used in strong motion inversion. [39] As the waveforms in Figure 15 indicate, increasing fractal dimension (i.e., more short-scale variability) results in more complex wave trains, while their low-frequency character remains similar. The differences in these waveforms and their peak velocities are large enough to matter in slip inversions using near-source recordings, and estimated slip distributions for well recorded earthquakes should reflect these differences. We also observe that velocity spectra (vertically shifted in Figure 15, right column) exhibit differences for all four fractal dimensions, but their frequency decay (/ w
1) remains unchanged for variable D, consistent with the general observation of flat acceleration spectra. 5.2. Additional Data to Constrain Slip Complexity [40] Limitations in resolution and accuracy of slip inversions limit the accuracy of the measured fractal dimension and correlation length, particularly at high wave numbers. It is therefore important to identify other data that may be useful to constrain the nature of earthquake slip complexity. [41] Perhaps the most intuitive source of data are measurements of surface slip for large earthquakes. These onedimensional surface-slip distributions could be analyzed both deterministically (i.e., correlating measured surface slip with imaged slip at depth) and stochastically (i.e., measuring correlation lengths and fractal dimension), and

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hence could provide additional insight into earthquake source complexity. The disadvantages of such surface-slip measurements, however, are their highly irregular spatial sampling, their large uncertainties [Hough et al., 2000], and the fact that such data exist only for very large earthquakes. We suspect, therefore, such data may be only of limited use to study earthquake source complexity. [42] Analyzing the spatial distribution of microseismicity is another possible approach for studying the spatial complexity of earthquake slip and stress. The spatial distribution of a values and b values, as imaged by Wiemer and Katsumata [1999], for example, could help to illuminate the spatial complexity of stress drop [Frankel, 1991]. Of course, microearthquake locations are also subject to uncertainties, so it would be most useful to work with a catalog of high-precision earthquake locations (e.g., D. P. Schaff et al., A high resolution image of Calaveras fault seismicity, submitted to Journal of Geophysical Research, 2002.), Such data are only available in very well-instrumented regions (e.g., California, Japan, and Taiwan), but could potentially help to constrain earthquake source complexity, particularly in terms of the geometric fault plane complexities. 5.3. Implications of Correlation Lengths for Rupture Dynamics [43] Uniform correlation lengths for all magnitudes would lead to problems in accommodating the slip (seismic moment) present in large earthquakes. If correlation lengths were about constant over a wide-magnitude range, a large earthquake would be comprised of many small, localized zones in which almost all the moment has to be released. These areas would have very large stress drops, yet would have to rupture in isolation from the surrounding high-slip areas in order to maintain short correlation lengths. It would be difficult to support such behavior from a rupture dynamics viewpoint where neighboring points of the fault interact strongly with each other, and hence are unlikely to allow isolated rupture zones of large stress drops. These arguments lead us to believe that the increasing correlation length for larger earthquakes is not an artifact due to the fault slip inversions (due to coarser fault discretization and longer periods for larger earthquakes), but rather a real property of the earthquake source. [44] One could argue that the observed correlation length scaling is merely determined by the overall extent of the source (geometry), and that at smaller length scales the slip function is purely self-affine (fractal) with a fractal dimension D arising from scale-independent rupture dynamics. We take an alternative view that the correlation length of the two-dimensional slip distribution of an earthquake is governed by length scales in the rupture process other than the overall source extent and that these may not be purely fractal. An earthquake will contain length scales spanning many orders of magnitudes, from the grain-size scale to total fault length. Length scales of juxtaposed rock units are on the order of 102 – 105 m [Warner et al., 1994; Holliger and Levander, 1992]. Geometric irregularities (fault bends, jogs, offsets) occur at lengths scales of 102 – 104 m and smaller. These ‘‘static’’ length scales will influence the characteristic lengths during an earthquake. It is also likely that the stress conditions on the rupture plane prior to and during the earthquake significantly influence the final

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MAI AND BEROZA: RANDOM FIELD MODEL FOR EARTHQUAKE SLIP

characteristic scales, leading to the observed scaling of the correlation lengths with source dimensions (seismic moment). [45] Moreover, the dynamics of earthquake rupture will also affect the correlation distances of the final slip distribution. If an earthquake encounters a strong asperity early in the rupture, the seismic load [Andrews, 1985] may not be sufficiently large to break the asperity, and the rupture is arrested early, resulting in a small earthquake with a short correlation distance. If the earthquake manages to break that asperity, its size and moment would grow, and the rupture would have a large area of slip. This scenario would result in a longer correlation distance for a somewhat bigger earthquake. Similarly, late into the rupture (when the seismic load is larger), stress drops may become large locally, and the rupture may become more difficult to stop and hence run to longer distances with significant slip, effectively increasing the correlation length of the final slip for large earthquakes.

6. Conclusions [46] We have developed an approach to characterize spatial complexity of earthquake slip imaged in finitesource rupture inversions. The characterization of slip heterogeneity as a spatial random field successfully captures the observed gradual roll-off of the wave number spectra as well as the directional dependence of the correlation lengths. In the case of slip being fractal, we find no indication that the fractal dimension, D, depends on other source parameters, and we conclude that D is independent of seismic moment. For the fractal model, however, the corner wave number kc (representing an ‘‘outer scale length’’) decreases with increasing source dimension. In contrast to the constant stress-drop model (D = 2), the observation D > 2 may imply size-dependent stress drop for subevents within an, earthquake incorrect mapping of variability in rupture speed and/or risetime into a more heterogeneous slip distribution, or extension of geometric fault trace complexity to depth. [47] Our preferred model is the von Karman ACF for which the correlation lengths a, ax, and az increase with increasing earthquake size. Independent studies [Somerville et al., 1999; Beresenv and Atkinson, 2001] are consistent with the increase of characteristic scale lengths with increasing magnitude that we have found. Simplified scaling relations imply that correlation lengths scale linearly with source dimension, in agreement with [Beresenv and Atkinson, 2001]. [48] The spatial random field model for earthquake slip can be used to generate scenario earthquakes for strong motion prediction. We calculate the source dimensions using scaling relations [e.g., Mai and Beroza, 2000], and generate spatially variable slip distributions using the spectral synthesis method [Pardo-Iguzquiza and Chica-Olmo, 1993]. It is important to note, however, that each realization of heterogeneous earthquake slip is subject to the condition that the static strain energy associated with the slip distribution remains finite (Appendix A). This condition ensures that stresses cannot become infinite (though locally they may become very large). Strong motion synthetics computed from these simulated slip distributions, assuming

simple kinematic rupture parameters (constant rupture velocity, constant risetime, boxcar source time function) show the utility of the model for strong motion prediction [Mai and Beroza, 2002]. So far, we have considered only heterogeneity in earthquake slip. Future work will address the question of how to obtain physically realistic distributions of risetime and rupture velocity, based on simulated slip (stress) distributions [Mai et al., 2001; Guatteri et al., 2002].

Appendix A [49 ] The wave number spectrum for heterogeneous source models cannot be arbitrarily flat because the stresses associated with the resulting slip distribution will become infinite. Andrews [1981] points out that individual grains of rock may support shear stress of several kilobars, and that stress fluctuations of that order may occur on short scales as individual grains fracture or slide over each other. Yet, average stress drops of earthquakes are observed to be on the order of 100 bars or less. Hence, stress fluctuations at the kilometer scale are much smaller. This conditions can be reconciled in terms of the static self-energy, Es, that has to remain finite for any slip distribution characterized by the correlation functions given in (1). [50] The change in static elastic energy during an earthquake is given as [Aki and Richards, 1980] Z Z  E¼


 1 t0 ðXÞ þ tðXÞ DðXÞdX 2

ðA1Þ

where t0(X) is the initial shear stress on the fault plane (generally unknown), t(X) and D(X) are the stress change and slip during the earthquake, respectively; X denotes the position on the fault plane. The static self-energy is given by the second term in (A1) Es ¼


1 2

Z Z

tðXÞDðXÞdX:

ðA2Þ

which can be expressed in the Fourier domain as [Andrews, 1980a] E s ðk Þ ¼


1 2

Z Z tðkÞDðkÞ dkx dkz

ðA3Þ

where * denotes complex conjugate. In the wave number domain, slip D(k) and stress t(k) are related by a static stiffness function, K(k), as [Andrews, 1980b] tðkÞ ¼
K ðkÞDðkÞ

ðA4Þ

where K ðk Þ ¼


  m 2ðl þ mÞ 2 kx þ kz2 2k l þ 2m

ðA5Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with k ¼ kx2 þ kz2 . For typical crustal rocks, l m, and (A5) simplifies to K ðkÞ ¼
2km 43 kx2 þ kz2
12 m k. Using (A3) and (A4) yields E s ðk Þ ¼

1 2

Z Z K ðkÞ DðkÞDðkÞ dkx dkz

ðA6Þ

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MAI AND BEROZA: RANDOM FIELD MODEL FOR EARTHQUAKE SLIP

which can be restated as Es ðkÞ ¼

1 2

Z Z K ðkÞPd ðkÞdkx dkz ;

ðA7Þ

with Pd(k) being the two-dimensional power spectrum of the slip distribution D(X). [51] The condition of finite static self-energy constrains the spectral decay of the slip spectrum at high wave number. To examine this condition for the spatial random field models in (1), we use the approximation that K(k)
12mk, and assume that ax2kx2 + az2kz2 1, reflecting the fact that the stresses are likely to become infinite only at small scales. Without loss of generality, we assume isotropy (ax az = a). The following discussion can be further simplified by using polar coordinates in k space, for which (A7) becomes E s ðk r Þ ¼

1 2

for all kr , and strain energy remains finite. The second case (H = 12) corresponds to the exponential ACF, and the static strain energy can be rewritten as E s ðk r Þ


A.1. Gaussian ACF [52] The PSD for the Gaussian model is given in (1); using polar coordinates and assuming isotropy, this becomes Pd ðkr Þ ¼

a2
1a2 k2r e 4 : 2

ðA9Þ

Inserting this expression into (A8) yields Z pma2 1 2 2 k2r e
4a kr dkr 4    pm c 1 1 2 2 kr e
4a kr þ Erfc akr
2 a 2

E s ðk r Þ


ðA10Þ

with c 3.54. This expression converges for all kr , and the condition of finite static strain energy is satisfied. A.2. von Karman ACF [53] Assuming isotropy and using polar coordinates, the PSD for the von Karman correlation function in (1) becomes

Z

1 dkr : kr

ðA13Þ

A.3. Fractal Model [54] We rewrite the PSD Pd(k) for the fractal model in (1) using polar coordinates as 1 Pd ðkr Þ /  4
D : k2r

ðA8Þ

Static strain energy will remain finite if the integral in (A8) converges for all kr.

pm 2a

This integral diverges as the logarithm of the wave number at the upper limit of kr . For case (3) with H < 12, the integral in (A12) diverges as a power law. There is infinite strain energy in the short wavelength for H  12, and hence Hurst exponents H  0.5 are not admissible to simulate heterogeneous slip distributions.

Z K ðkr ÞPd ðkr Þ2p k r dkr ;

10 - 19

ðA14Þ

Using (A8), the static self-energy is given as Es ðkr Þ /

pm 2

Z

1 ðkr Þ6
2D

dkr :

ðA15Þ

This integral expression is unbounded for slip distributions with fractal dimension D 2.5; for D = 2.5, the integral diverges as the logarithm of the wave number at the upper limit of kr , for D > 2.5, it diverges as a power law. Fractal dimensions of D < 2.5 (as found in the analysis of finitesource models) or D = 2 (as in the ‘‘k-square’’ model [Herrero and Bernard, 1994]) are admissible to simulate slip distributions. [55] Acknowledgments. We are indebted to numerous researchers who sent us their slip maps electronically. We want to particularly thank D. Wald for maintaining the finite-fault source model repository. T. Mukerji provided initial codes for the spectral synthesis method. We enjoyed discussion with J. Andrews, P. Guatteri, P. Segall, N. Sleep, and P. Spudich. Joe Andrews’ skepticism stimulated our thinking, and his careful review of an earlier version of the manuscript improved the work. Constructive comments by R. Abercrombie, M. Bouchon, and an anonymous reviewer helped to improve the manuscript. This research was supported by the US – Japan Cooperative Research in Urban Earthquake Disaster Mitigation project (NSF 98-36) grant CMS-9821096.

References 2

Pd ðkr Þ ¼ 

a

1þ

Hþ1 a2 k2r

:

ðA11Þ

Inserting this expression into (A8) we find that Z Es ðkr Þ
cH 2

1 dkr : k2H r

ðA12Þ

mH 1 where cH ¼ 2p KH ð0Þ a2H . This integral expression for the static strain energy depends on the Hurst exponent, H, and three cases are of particular interest: (1) H > 0.5, (2) H = 0.5, and (3) H < 0.5. In the first case (H > 12), the integral converges

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G. C. Beroza, Department of Geophysics, Stanford University, Stanford, CA 94305-2215, USA. P. M. Mai, Institute of Geophysics, ETH Hoenggerberg, CH-8093 Zurich, Switzerland. ([email protected])